Question: Factor the following expression: $9$ $x^2+$ $22$ $x$ $-15$
Explanation: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(9)}{(-15)} &=& -135 \\ {a} + {b} &=& & & {22} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-135$ and add them together. Remember, since $-135$ is negative, one of the factors must be negative. The factors that add up to ${22}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-5}$ and ${b}$ is ${27}$ $ \begin{eqnarray} {ab} &=& ({-5})({27}) &=& -135 \\ {a} + {b} &=& {-5} + {27} &=& 22 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {9}x^2 {-5}x +{27}x {-15} $ Group the terms so that there is a common factor in each group: $ ({9}x^2 {-5}x) + ({27}x {-15}) $ Factor out the common factors: $ x(9x - 5) + 3(9x - 5) $ Notice how $(9x - 5)$ has become a common factor. Factor this out to find the answer. $(9x - 5)(x + 3)$